15edt: Difference between revisions
No edit summary |
mNo edit summary |
||
| (One intermediate revision by one other user not shown) | |||
| Line 3: | Line 3: | ||
== Theory == | == Theory == | ||
15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1|5]] and [[13/1|13]] closely in tune, but does not do so well for [[11/1|11]], which is quite sharp. The main appeal of 15edt is that it allows for strong tritave equivalency, while supporting more conventional harmony. It achieves this with fantastic approximation of the [[4/1|4th harmonic]], and terrible approximation of the [[2/1|octave]]. In other words; 3:4:5 is available, but 4:5:6 is not. Like the octave, the [[7/1|7th harmonic]] is about halfway between steps, so 6:7:8 is well approximated, but not 4: | 15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1|5]] and [[13/1|13]] closely in tune, but does not do so well for [[11/1|11]], which is quite sharp. The main appeal of 15edt is that it allows for strong tritave equivalency, while supporting more conventional harmony. It achieves this with fantastic approximation of the [[4/1|4th harmonic]], and terrible approximation of the [[2/1|octave]]. In other words; 3:4:5 is available, but 4:5:6 is not. Like the octave, the [[7/1|7th harmonic]] is about halfway between steps, so 6:7:8 is well approximated, but not 4:5:7. It also tempers out the syntonic comma, [[81/80]], in the 3.4.5 subgroup, as the major third is three perfect fourths below a tritave. As a 3.5.13-[[subgroup]] system, it tempers out [[2197/2187]] and [[3159/3125]], and if these commas are added, 15edt is related to the 2.3.5.13-subgroup temperament 19 & 123, which has a mapping {{mapping| 1 0 0 0 | 0 15 22 35 }}, where the generator, an approximate 27/25, has a [[POTE tuning]] of 126.773, very close to 15edt. | ||
Using the patent val, it tempers out [[375/343]] and [[6561/6125]] in the 7-limit; [[81/77]], [[125/121]], and [[363/343]] in the 11-limit; [[65/63]], [[169/165]], [[585/539]], and [[1287/1225]] in the 13-limit; [[51/49]], [[121/119]], [[125/119]], [[189/187]], and [[195/187]] in the 17-limit (no-twos subgroup). With the patent [[4/1|4]], it tempers out [[36/35]], [[64/63]], and 375/343 in the 3.4.5.7 subgroup; [[45/44]], [[80/77]], 81/77, and 363/343 in the 3.4.5.7.11 subgroup; [[52/49]], 65/63, [[65/64]], [[143/140]], and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, [[52/51]], [[85/84]], and 121/119 in the 3.4.5.7.11.13.17 subgroup ( that 15edt treated this way is essentially a retuning of [[19ed4]]). The [[k*N subgroups|2*15 subgroup]] of 15edt is 3.4.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for [[30edt]]. | |||
15edt is also associated with [[tempering out]] the mowgli comma, {{monzo| 0 22 -15 }} in the [[5-limit]], which fixes [[5/3]] to 7\15edt; in an octave context, this temperament is supported by [[19edo]] but has an [[optimal patent val]] of [[303edo]]. | |||
=== Harmonics === | === Harmonics === | ||